RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Preparing for a smallpox bioterrorist attack: pulse vaccination
as an optimal strategy
Zvia Agur1, Karen Marron1, Hanita Shai1, Yehuda L. Danon2
1. Institute for Medical BioMathematics (IMBM); 10 Hate’ena St.; P.O.B. 282,
60991; Bene Ataroth, Israel
2. Kipper Institute of Immunology; Schneider Children Medical Center; FMRC;
Petach-Tikva, 49202; Israel
Events of recent years have significantly increased our awareness of the
potential threat of a bioterrorist attack, and the smallpox variola virus has been identified
as an “eligible candidate” for a biological warfare agent. Epidemiological mathematical
modeling has long been recognized as a crucial tool for assessing the repercussions of
such a viral outbreak, as well as for comparing possible response strategies. Several
studies have applied mathematical models with the hope of identifying an optimal
vaccination strategy in case of an attack [1–4]. However, these particular studies do not
produce clear-cut recommendations, upon which authorities can rely when formulating
policies. In fact, conclusions of certain studies contradict those of others.
This inconsistency may stem from simulation results’ extreme sensitivity to the
value assumed for the disease’s basic reproduction rate (R0) [5], i.e. the number of
secondary infections that an infectious individual is expected to produce if introduced
into an entirely susceptible population. Because smallpox was eradicated in 1979, there
is a lack of current epidemiological data on the disease, and R0 must be estimated.
However, as this value is dependent both on biological factors and on population
1
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
dynamics, it is extremely difficult to assess prior to an actual epidemic. Each of the
above-mentioned studies assumes a different value for R0, which may account for their
contradictory results: a response strategy found to be effective in the case of a low
disease transmission rate may fall short when faced with a higher transmission rate.
Furthermore, there is a fundamental limitation that compromises the practical
value of theoretical smallpox studies to date: they examine response strategies that
address only an initial epidemic, without considering long-term effects of the introduction
of the disease into a contemporary population. Theory indicates that smallpox, like many
other contagious diseases, is characterized by decaying oscillatory dynamics, i.e.
periodic epidemics of decreasing magnitude [6]. Thus, while a given vaccination strategy
may succeed in curbing an outbreak immediately following the release of the virus, it
may not be able to prevent additional epidemics several years down the line (see Figure
1). Clearly, an efficient follow-up strategy is no less important than initial crisis aversion.
Nevertheless, most smallpox studies either fail to mention such a strategy, or simply
assume that the entire population will continue to be vaccinated in the years following
the outbreak. This type of continuous vaccination policy would be far from optimal, as
vaccinia, the smallpox vaccine, is known to cause serious, and even fatal, side effects.
Additionally, the vaccine carries quite a few contra-indications: for example, it is not
recommended to administer it to young children [6]. Due to these significant constraints,
it becomes evident that a general idea for a long term vaccination strategy is insufficient;
this strategy must be rigorously defined. Theory implies that efficient control of
phenomena such as smallpox transmission, which display clear periodicity, should
involve a periodic strategy with a period different from that of the phenomenon being
counteracted [7, 8].
Based on these guidelines, we examine a vaccination strategy that has not yet
been considered in the context of response to a smallpox bioterrorist attack: pulse
2
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
vaccination, i. e. periodic vaccination of certain percentages of the susceptible
population following attack. The pulse strategy has previously been proposed as an
efficient method for measles eradication [9, 10]. In addition to its theoretically proven
efficacy, this strategy also enables flexibility: the number of people vaccinated and the
duration of time between vaccination campaigns (inter-vaccination interval) can be
varied according to arising circumstances. This flexibility is crucial when dealing with a
situation as sensitive as a smallpox bioterrorist attack, with so many unknown and
unpredictable factors, e.g. the disease’s actual basic reproduction rate, the population’s
response to an outbreak and willingness to comply with vaccination campaigns,
available vaccine stockpile, vaccine efficacy, etc.
In this study, we compare the pulse vaccination strategy to two known smallpox
response strategies: (i) preliminary vaccination, i.e. vaccination of a certain percentage
of the population prior to the attack; (ii) one-time mass vaccination, in which most of the
population is vaccinated immediately following the attack. Additional vaccination
strategies that have been considered for smallpox protection are trace and ring
vaccination, in which vaccination is limited to close contacts of infected individuals.
Though these approaches are potentially effective in large countries [11], they are not
considered feasible in small countries with high population density such as Israel, and
therefore we did not include them in our study.
To compare the performance of the different vaccination strategies under a wide
range of conditions, we adapted the SEIR epidemiological model [6] to describe the
effects of a smallpox bioterrorist attack on the Israeli population over a period of ten
years. We simulate and compare the different vaccination strategies under the
constraints of low compliance, insufficient vaccine stockpile, contra-indications, sideeffects, cost of vaccination, etc. In order to obtain robust conclusions, we evaluate the
3
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
performance of each studied vaccination strategy over a wide range of potential basic
reproduction rate (R0) values (1.5 ≤ R0 ≤ 30).
It is important to note that though these simulations were carried out with
smallpox in mind, the results are applicable to almost any infectious disease.
Mathematical Modeling
We adapted a classic SEIR model for the transmission of an infectious disease in
a spatially heterogeneous population [6]. We assume that the smallpox virus is
introduced into the Israeli population by one infective carrier. The model takes into
account three stages of infection [12], assuming that each stage has a constant duration
(d1, d2 and d3 in Table 1 respectively): (1) infected, noninfective and vaccine sensitive;
(2) infected, noninfective, vaccine-insensitive; (3) infected and infective. We define the
basic transmission rate as: β =
R0
, where N is the initial population size. A certain
Nd 3
fraction of infective cases die of the disease, and the remaining fraction eventually
recovers, becoming immune to reinfection. The model does not consider loss of
immunity following vaccination or recovery from the disease [13].
We examine three vaccination policies as follows: (i) Preliminary vaccination:
Different percentages of the population are vaccinated prior to attack, from 0 to 100% in
increments of 5%; (ii) One-time mass vaccination: We define a “maximum vaccination
capacity”, i.e. the maximum number of people that can be vaccinated per day (see Table
1). The percentage of capacity utilized (PCU) is varied between 0 and 100% in
increments of 5%, and the duration of the vaccination campaign is varied between 1 and
15 days in increments of 1 day; (iii) Pulse vaccination: PCU is varied from 0 to 100% in
increments of 5%, and the interval between cycles is varied from 50 to 2000 days in
4
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
increments of 50 days. Due to computational considerations, the duration of periodic
vaccination campaigns is set as constant (3 days).
When applying one-time mass vaccination or pulse vaccination strategies,
priority is given to individuals who have been exposed to the disease but can still be
effectively vaccinated.
Strategy assessment: the “cost” of an outbreak
Theoretically, continuous vaccination of the entire population would be a
foolproof means of preventing disease outbreak. However, as previously mentioned, this
would not be the optimal strategy, due to the constraints imposed by the vaccinia
vaccine [12]. In addition, it would probably not be reasonable to expect full compliance,
especially over a long period of time after the initial outbreak. Therefore, an optimal
strategy should not only prevent disease outbreak, but should allow as few individuals as
possible to be vaccinated.
Vaccinating fewer individuals will not only prevent side effects, but will be less
disruptive to routine. Thus, in order to assess the efficacy of a particular strategy, we
determine a “cost” which takes into account the following factors: (i) number of deaths by
infection (Dinf ); (ii) number of infections (I); (iii) number of people vaccinated (Nvac); (iv)
number of deaths resulting from vaccination (Dvac); (v) non-fatal side effects of
vaccination (E); (vi) a “stress factor”: number of post-attack vaccinations performed per
day (S).
Each factor is attributed a “weight”, and the cost C is calculated as follows:
C = (WDinf +
W
W
I + N vac + E + 2WDvac )(1 + Se −7 ).
2
10
5
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Where W is an “anchor weight” which represents the ratio between the weight of
death by infection and the number of vaccinees. This value can be interpreted as the
number of individuals a country is willing to vaccinate in order to save one life. Clearly,
the resulting cost is closely dependent on the value of these weights. Therefore, to
obtain robust results, we run simulations over a wide range of “anchor weights” (range:
1-10,000).
Model parameters are presented in Table 1, where demographic parameters
apply to the Israeli population. Additional parameter values such as disease phase
duration, etc. are taken from classic references. Clearly, disease and vaccine fatality
rates are extremely influential when determining the tradeoff between the number of
vaccinees and the number of deaths by disease. The model’s possible sensitivity to
these parameters is accounted for by varying the “anchor weight” applied when
computing the cost (see above). As previously mentioned, a crucial parameter for
assessing the effects of an outbreak is R0. However, this parameter is also extremely
difficult to estimate, as it is dependent on dynamic factors such as socioeconomic
conditions [6, 12]. The basic transmission rate of smallpox has been estimated between
3 and 10 [14, 15], and values implemented in studies vary between 1.5 and 15 (e.g. [2,
3]). Instead of assuming a single value for R0, we run simulations over a wide range of
values, which vary between 1.5 and 30.
Results
As previously described, we simulated and compared various applications of
preliminary, one-time mass vaccination and pulse vaccination strategies. Each simulated
strategy design produced a final cost, a low cost signifying an efficient strategy, and a
relatively high cost signifying an inefficient strategy. Figure 2 presents the distribution of
6
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
costs attained by each strategy simulated, for several values of R0. Different costs for the
same strategy are achieved by varying strategy design (e.g. a preliminary vaccination
strategy in which 75% of the population is vaccinated may produce a lower cost than a
preliminary strategy in which only 20% of the population is vaccinated). Figure 2
indicates that for R0 ≤1.5, all three vaccination strategies tested can potentially be
efficient: if properly designed, they can reduce “cost” to 0. However, for R0 > 1.5, pulse
vaccination is the only strategy that succeeds in minimizing cost. This result holds true
even for R0 > 10, i.e. beyond the range of values that have been estimated. The same
qualitative results are obtained even when the “anchor weight” is varied, provided that
the weight chosen indicates vaccination to be preferable over non-vaccination (Figure 3).
Pulse Vaccination: finding the optimal design
Though pulse vaccination can potentially be an effective strategy, inappropriate
strategy design is likely to produce high costs (see Figure 2). Therefore it is crucial to
identify the optimal strategy design, in terms of PCU and the interval between “pulses”.
Figure 4 presents graphs of cost as a function of PCU and interval between pulses for
several values of R0.
As is evident from these graphs, for each R0 value there is a “zone” (shaded
areas) in which costs are low, and outside of which costs rise suddenly. This zone is
defined by a diagonal contour, which means that there is a critical ratio of PCU to
interval between pulses above which the pulse strategy will effectively reduce costs, and
below which the strategy is inefficient. This critical ratio increases in direct relation to R0
(Figure 5).
Discussion
7
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Our study suggests that compared to one-time mass vaccination and preliminary
vaccination, pulse vaccination, i.e. multiple vaccination campaigns with precisely
calculated inter-vaccination intervals, is by far the most promising strategy for ensuring
long-term population protection against smallpox. Simulations show that while one-time
mass vaccination and preliminary vaccination may be effective vaccination strategies for
a virus with a low basic reproduction rate (R0 ≤ 1.5), a pulse vaccination strategy can be
designed so as to prevent disease outbreak for a broad range of R0 values, and over a
long period of time. We analyzed the effect of the two crucial pulse strategy parameters namely the coverage level and the inter-vaccination intervals - on the final “cost” of the
outbreak, and reached the conclusion that there exists a critical ratio between the
percent of vaccination capacity utilized and the interval between vaccination campaigns
above which the strategy will be effective, and below which the “cost” of attack will rise
suddenly.
These results may carry significant implications for countries planning a response
to a smallpox bioterrorist attack, or to an outbreak of any disease for which a vaccine
exists, but about which little else is known. They indicate that given the realistic range of
the disease’s basic reproduction rate and the estimated compliance level, authorities can
design a detailed, optimal vaccination plan for years to come. This plan would be
extremely flexible and robust: if unexpected circumstances should arise, e.g. insufficient
stockpiles or low compliance, maximum protection could still be ensured by modifying
the interval between vaccination campaigns, or the number of people vaccinated at each
campaign. Additionally, the pulse strategy would enable authorities to test their facilities,
e.g. by initially vaccinating first responders, and only later the general public. This
preparation would allow them to draw conclusions for further campaigns, increasing
overall efficiency.
8
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Clearly, for both bureaucratic and psychological reasons, the government of a
country may prefer to provide a mass or preliminary vaccination policy. The pulse
strategy would not have to replace either of these strategies. Rather, it could be used as
a backup strategy, and as a long term policy, which would probably become increasingly
necessary, as compliance may decrease after initial panic subsides. The critical ratio
between the percent capacity utilized and the interval between pulses offers authorities
quite a bit of leeway when deciding upon the timing and the magnitude of these
campaigns.
Acknowledgments
This work was supported in part by the Chai Foundation, and in part by the Israeli
Defense Forces.
References
1. Meltzer MI., Damon I, LeDuc JW, Millar JD. Modeling potential responses to
smallpox as a bioterrorist weapon. Emerg Infect Dis. 2001;7:959–969.
2. Kaplan EH, Craft DL, Wein LM. Emergency response to a smallpox attack: The
case for mass vaccination. Proc Natl Acad Sci USA. 2002;99:10935–10940.
3. Bozette SA, Boer R, Bhatnagar V, Brower JL, Keeler E, Morton SC, et al. A
model for a smallpox-vaccination policy. N Engl J Med. 2002;348:416–425.
4. Halloran ME, Longini IM, Nizham A, Yang Y. Containing bioterrorist smallpox.
Science. 2002;298:1428–1432.
5. Ferguson NM, Keeling MJ, Edmunds WJ, Gani R, Grenfell BT, Anderson RM et
al. Nature. Planning for smallpox outbreaks. 2003;425:681–685.
9
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
6. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control.
Oxford: Oxford Univ. Press; 1991.
7. Agur, Z. Randomness, synchrony and population persistence. J Theor Biol.
1985;112:677–693.
8. Agur, Z. Resonance and anti-resonance in the design of chemotherapeutic
protocols. J Theor Med. 1998;1:237–245.
9. Agur Z, Cojocaru L, Mazor G, Anderson RM, Danon YL. Pulse mass measles
vaccination across age cohorts. Proc Natl Acad Sci USA. 1993;90:11698–11702.
10. Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic
model. Bull Math Biol. 1998;60(6):1123–1148.
11. Fraser C, Riley S, Anderson RM, Ferguson N. Factors that make an infectious
disease outbreak controllable. PNAS. 2004;101:6146–6151.
12. Fenner F, Henderson DA, Arita I, Jezek Z, Ladnyi ID. Smallpox and its
Eradication. Geneva: World Health Organization; 1988.
13. Eichner M. Analysis of historical data suggests long-lasting protective effects of
smallpox vaccination. Am J Epidemiol. 2003;158:717–723.
14. Gani R, Leach S. Transmission potential of smallpox in contemporary
populations. Nature. 2001;414:748–751.
15. Eichner M, Dietz K. Transmission potential of smallpox: estimates based on
detailed data from an outbreak. Am J Epidemiol. 2003;158:110–117.
10
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Table 1: Model Parameters
Parameter
Value
Unit
Initial susceptible population
6439200
individuals
Initial infective population
1
individual
Birth rate
9e-4
Death rate
9e-4
First stage disease duration (d1)
4
days
[12]
Second stage disease duration (d2)
13
days
[12]
Third stage disease duration (d3)
21
days
[12]
Disease fatality rate
0.3
[12]
Vaccine fatality rate
10-6
[12]
Rate of non-fatal vaccination side effects
0.001
[12]
Maximum daily vaccination capacity
1.8e6
individuals
Duration of pulse vaccination campaign
3
days
11
Reference
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Figure 1: Simulation results of the adapted SEIR model: Number of infected individuals
over time when disease is introduced into the Israeli population by one infective carrier.
Thick line: natural disease dynamics (no vaccination strategy is implemented). Thin line:
Disease dynamics after implementation of one-time mass vaccination strategy
population is vaccinated at maximum capacity for three days following the attack).
12
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Figure 2: Costs produced by each simulated vaccination strategy for several values of
R0. Strategies tested: preliminary vaccination (squares), one-time mass vaccination
(triangles) and pulse vaccination (circles). Different costs for the same strategy are
obtained by varying strategy design.
13
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Figure 3: Costs produced by each simulated vaccination strategy for several values of
the anchor weight (W = 3; 10; 100; R0 is 3.5). As in Figure 2, strategies tested are
preliminary vaccination (squares), one-time mass vaccination (triangles) and pulse
vaccination (circles). Note that for W = 3, pulse vaccination is not more efficient than
other strategies, but at the same time non-vaccination is as efficient as vaccination. For
larger W values, pulse vaccination is more efficient.
14
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Figure 4: Pulse vaccination: cost as a function of percent vaccination capacity utilized
and interval (in days) between pulses, for several R0 values. There is a critical ratio
(diagonal contour) between the two, above which the strategy will be effective (shaded
area) and below which costs rise suddenly.
15
RISK ASSESSMENT AND RISK COMMUNICATION STRATEGIES IN
BIOTERRORISM PREPAREDNESS
Figure 5: There is a critical ratio of percent vaccination capacity utilized to interval
between pulses that must be exceeded in order for a pulse strategy to be effective. In
graph: critical ratio as a function of R0.
16